source file: mills2.txt Date: Fri, 20 Sep 1996 01:27:30 -0700 Subject: Re: duh? From: kollos@cavehill.dnet.co.uk (Jonathan Walker) Scott_Purman@nile.com (Wed, 18 Sep 96, Subject: duh?) said: > [C]an someone explain to me what the ratios are ratios of? > or maybe tell me the ratios of the 12 tones in a > traditional ET scale (or 13 tones, from C to C for > instance). I think I could pick it up from > there....thanks. In answer to Scott Purnham, and on the assumption that others new to the list may have similar queries, I offer here a brief explanation of ratios, JI and 12TET. Apologies to the great majority of list subscribers for cluttering up their mailboxes -- you can skip to the next message. What are they? -- The ratios of the frequencies of two pitches. 3/2 is a perfect fifth, 5/4 a just major third, 81/64 a Pythagorean major third, 81/80 the syntonic comma (difference between 5/4 and 81/64). To decipher the ratios, you must extract the prime factors in the numerator and denominator: e.g. 5/4 = 2^-2 * 5^1 81/80 = 2^-2 * 3^4 * 5^-1 "*" is multiplication, "^" is an exponent superscript sign (so 3^4 is three-to-the-power-of-four, i.e. 81). I don't know just how rusty your maths might be, so I'll remind you that the negative exponent x^-1 means 1/x, so 2^-2 is 1/4, while 2^-2 * 5^1 is 5/4. The intervals in JI Ear Trainer are all smaller than an octave, so the numerator of the ratios will always be greater than the denominator, but less than twice the denominator; i.e. the ratios fall between 1/1 (unison) and 2/1 (the octave). Since you will be familiar with the harmonic series, this might be the most convenient point of entry. We are not concerned here with absolute frequency, expressed in Hz, but only with frequency relations. Define the fundamental as 1, so the second harmonic, an octave higher has twice the frequency, thus 2/1; the third harmonic, an octave and a fifth higher, has three times the frequency, thus 3/1; the fourth harmonic, two octaves higher is 4/1, i.e. (2^2)/1; the fifth harmonic, two octaves and a (just) major third higher is 5/1. If a perfect fifth plus an octave is 3/1, how do you express a perfect fifth alone in ratio terms? Just as multiplying the frequency by 2 raises a pitch by an octave, so division by 2 lowers a pitch by an octave; thus 3/2 is 3/1 less an octave -- a perfect fifth above the fundamental. How do you express a (just) major third in ratio terms? Since 5/1 is a major third plus two octaves, divide 5 by 4 -- that is, by 2^2 -- to bring it down to a major third above the fundamental; 5/4 is therefore a major third. But I also mentioned a Pythagorean major third above; how is this derived? Multiplying a frequency by 3 is equivalent to raising a pitch by an octave and a fifth; now, say your fundamental is C, then 3 times the fundamental is a G, 3^2 or 9 times is a D, 3^3 or 27 times is an A, and 3^4 or 81 times is an E -- this is the 81st harmonic. In order to bring this E within the octave above the fundamental, we'll have to descend by six octaves, so we divide 81 by 2^6 or 64, hence the ratio 81/64 for the Pythagorean major third. How is the 81/64 third related to the 5/4 third? To find which is higher, express the two ratios in terms of a common denominator: thus we have 81/64 to compare with 80/64 -- this means that the Pythagorean third is higher. To express the difference in ratio terms, divide 81/64 by 5/4; this is equivalent to 81/64 * 4/5, which equals 81/80, i.e. the syntonic comma I mentioned above. You can now manipulate the ratios accordingly, without reference to the harmonic series. Such ratios were in any case discussed by music theorists long before the harmonic series and the frequency of sound-waves were understood: the Chinese took the relative lengths of bamboo pipes as their conceptual starting-point, Pythagoras the relative weights of hammers, and thereafter the divisions of a string became the most common means of conceptualising ratios (the monochord of mediaeval and Renaissance musicians was such a device -- it was never intended for performance). So far I've left "Pythagorean" and "just" undefined, but we now have the means to rectify this. A Pythagorean interval is expressible in terms of a ratio that includes only 2 and 3 (and their powers) in the numerator and denominator. A just interval will have a ratio that also includes 5 (and its powers) in the numerator and denominator. "Extended just intonation" includes prime numbers higher than 5. Partch's terminology is the most lucid for such purposes: Pythagorean intervals are 3-limit intervals, just intervals are 5-limit, while Partch used an 11-limit system. Note that each higher number limit includes the previous limit, so the 3-limit intervals form a proper subset of the 5-limit set; each sytem is, theoretically speaking, an infinite set of intervals (this doesn't conflict with the fact that one can be a subset of another). A quick test of the knowledge so far gained: express the major seventh in ratio terms, within the 5-limit system. Just as we can construct such an interval by adding a perfect fifth and a major third (in this case, just), so we multiply 3/2 and 5/4, to obtain 15/8. Finally, a word on the intervals of12-note equal temperament (12TET); these are irrational proportions, so we will have to abandon ratio terminology. The tuning system is constructed upon that proportion which, when multiplied by itself 12 times, equals 2. Thus we have to find an interval above 1/1, the starting pitch, twelve of which will make up an octave, i.e. 2/1. That interval is the twelfth root of 2 above the 1/1 starting point, the starting frequency multiplied by 2^(1/12) -- that is, two to the power of a twelfth, or the twelfth root of two. Thus the equal tempered semitone is the proportion (2^[1/12])/1. How is the 12TET perfect fifth expressed in such terms? Since this fifth is seven 12TET semitones above an initial pitch, we want the seventh power of the twelfth root of 2, i.e. (2^[1/12])^7 = 2^(7/12). How can this be compared to the 3/2 perfect fifth? Simply find the decimal value of each; you'll find the 3/2 is very slightly higher. The 12TET system was adopted as a tuning convenience in the late 18th/early 19th centuries in Europe because it provided a good approximation of the common 3-limit intervals, and a tolerable -- or at least recognizable -- approximation of the 5-limit intervals. An equivalent method for constructing 12TET is to find the difference between 12 fifths and 7 octaves (the Pythagorean comma), and remove the twelfth root of this difference from each fifth; from this the "circle of fifths" is artificially derived. Each 12TET fifth will therefore be 3/2 divided by the twelfth root of the Pythagorean comma; I'll leave you to prove for yourself that this is equivalent to 2^(7/12). -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 20 Sep 1996 12:20 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07871; Fri, 20 Sep 1996 12:21:58 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA06325 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id DAA24800; Fri, 20 Sep 1996 03:21:55 -0700 Date: Fri, 20 Sep 1996 03:21:55 -0700 Message-Id: <009A8A7640C705C0.F61C@vbv40.ezh.nl> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu